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A General Class of Functionals for Certifying Quantum Incompatibility

Kuan-Yi Lee, Jhen-Dong Lin, Adam Miranowicz, Yueh-Nan Chen

Abstract

Quantum steering, measurement incompatibility, and instrument incompatibility have recently been recognized as unified manifestations of quantum incompatibility. Building on this perspective, we develop a general framework for constructing optimization-free, nonlinear incompatibility witnesses based on convex functionals, valid in arbitrary dimensions. We prove that these witnesses are nontrivial precisely when the underlying functional is non-affine on extremal points (e.g., pure states for ensembles). For pure bipartite states, the witnesses yield lower bounds on entanglement measures, thereby outperforming most linear steering inequalities in the pure-state regime. Moreover, the construction extends in full generality to certify measurement and instrument incompatibility, where the witnesses act as genuine incompatibility monotones. We demonstrate the versatility of our approach with two operationally relevant functionals: the Wigner-Yanase skew information and an $\ell_{2}$-type coherence functional.

A General Class of Functionals for Certifying Quantum Incompatibility

Abstract

Quantum steering, measurement incompatibility, and instrument incompatibility have recently been recognized as unified manifestations of quantum incompatibility. Building on this perspective, we develop a general framework for constructing optimization-free, nonlinear incompatibility witnesses based on convex functionals, valid in arbitrary dimensions. We prove that these witnesses are nontrivial precisely when the underlying functional is non-affine on extremal points (e.g., pure states for ensembles). For pure bipartite states, the witnesses yield lower bounds on entanglement measures, thereby outperforming most linear steering inequalities in the pure-state regime. Moreover, the construction extends in full generality to certify measurement and instrument incompatibility, where the witnesses act as genuine incompatibility monotones. We demonstrate the versatility of our approach with two operationally relevant functionals: the Wigner-Yanase skew information and an -type coherence functional.
Paper Structure (7 sections, 52 equations, 2 figures, 1 table)

This paper contains 7 sections, 52 equations, 2 figures, 1 table.

Table of Contents

  1. Derivation of a class of incompatibility witnesses
  2. Proof of Result 2: The Nontriviality of Witnesses
  3. Examples of non-trivial inequalities for state assemblages
  4. Proof of Result 3: The Witnesses Lower bounds a full entanglement measure
  5. Proof of Result 4: A valid incompatibility monotone
  6. Example of qubit Pauli measurements for pure entangled state input
  7. Instrument assemblage and channel steering under amplitude damping channel and noisy Pauli measurements

Figures (2)

  • Figure 1: Schematic illustration of compatibility tasks.
  • Figure 2: The violation degrees of (a) $\mathcal{S}_{I}(\theta, w)$ and (b) $\mathcal{S}_{N}(\theta, w)$, where the black-dashed curves represent the boundaries of zero violation. The largest violations appear at $\theta = \pi /4$ are marked as blue curves; at these slices, $\mathcal{S}_I|_{\theta=\pi/4}=\mathcal{M}_I(w)$ and $\mathcal{S}_N|_{\theta=\pi/4}=\mathcal{M}_N(w)$ and both monotonically decrease when $w$ increases. One can observe that the $\mathcal{M}_I$ is more sensitive to noise and vanishes as $w\approx 0.21$, while the $\mathcal{M}_N$ meets the threshold of compatible measurement, i.e., $w=1- 1/\sqrt{2}\approx 0.29$. The violation degrees of (c) $\mathcal{M}_I(\gamma,w)$ and (d) $\mathcal{M}_N(\gamma,w)$. Analogously, when $w=0$, $\mathcal{M}_I|_{w=0}$ and $\mathcal{M}_N|_{w=0}$ reduce to $\mathcal{I}_I(\gamma)$ and $\mathcal{I}_N(\gamma)$, respectively, indicating the instrument incompatibility (see the red curves in each figures). Additionally, let $\gamma = 0.5$, the minimum dilation maps the input state $\ket{1}$ to a maximally entangled state, reflecting the measurement incompatibility $\mathcal{M}_I(\boldsymbol{M})$ and $\mathcal{M}_N(\boldsymbol{M})$.

Theorems & Definitions (1)

  • proof